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Answer:
a. f(x) = (4x +5)(x -3)
b. x-intercepts: {-1.25, +3}
c. f(x) goes to +infinity as x goes to ±infinity
d. plot the x-intercepts, the vertex, and a few other points; draw a smooth curve through them
Step-by-step explanation:
a. Here, you're looking for factors of (4)(-15) = -60 that have a sum of -7. Those factors will be -12 and +5, so we can factor the function as ...
f(x) = (4x -12)(4x +5)/4
f(x) = (x -3)(4x +5)
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b. The x-intercepts are the values of x that make the factors zero:
x -3 = 0 ⇒ x = 3 . . . . (add 3)
4x +5 = 0 ⇒ x = -5/4 . . . . (add -5, divide by 4)
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c. As with all even-degree polynomials with positive leading coefficients, the end behavior of f(x) matches that of |x|:
as x → ∞, f(x) → ∞
as x → -∞, f(x) → ∞
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d. The vertex of the function will be located halfway between the x-intercepts, at x = (3-5/4)/2 = (7/4)/2 = 7/8. The value of f(x) there is ...
f(7/8) = (4(7/8) +5)(7/8 -3) = -289/16 = -18.0625
Since the leading coefficient is 4, the normal x^2 function behavior is stretched vertically by a factor of 4. So, additional points would be
x = 7/8 ±1, f(x) = -18.0625 +4·1 = -14.0625
x = 7/8 ±2, f(x) = -18.0625 +4·2^2 = -2.0625
The answers of part B locate two points on the graph, and help find the vertex. The answer of part C confirms the graph as opening upward.
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Additional comment
I find a graphing calculator to be very helpful finding the roots and making the graph.
